Discrete Mathematics and its Applications 9/17/2018 University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter Slides adapted from Michael P. Frank’s Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. The instructor can bring several modules to each lecture with him, to make sure he has enough material to fill the lecture, or in case he wants to preview or review slides from upcoming or recent past lectures. 9/17/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Rosen 5th ed., chs. 8-9 ~44 slides (more later), ~3 lectures Module #22: Graph Theory Rosen 5th ed., chs. 8-9 ~44 slides (more later), ~3 lectures 9/17/2018 (c)2001-2003, Michael P. Frank

What are Graphs? Not General meaning in everyday math: A plot or chart of numerical data using a coordinate system. Technical meaning in discrete mathematics: A particular class of discrete structures (to be defined) that is useful for representing relations and has a convenient graphical representation. 9/17/2018 (c)2001-2003, Michael P. Frank

Applications of Graphs (At least) anything that can be modelled using 1- or 2-place relations. Apps in networking, scheduling, flow optimization, circuit design, path planning, search; genealogy, any kind of taxonomy, … Challenge: name a topic that graphs cannot model 9/17/2018 (c)2001-2003, Michael P. Frank

We shall introduce a number of different types of graphs, starting with undirected graphs: “simple” graphs multigraphs pseudographs 9/17/2018 (c)2001-2003, Michael P. Frank

Visual Representation of a Simple Graph Simple Graphs Correspond to symmetric, irreflexive binary relations R. A simple graph G=(V,E) consists of: a set V of vertices or nodes (V corresponds to the universe of the relation R), a set E of edges: unordered pairs of distinct elements u,v  V, such that uRv. Visual Representation of a Simple Graph 9/17/2018 (c)2001-2003, Michael P. Frank

Example of a Simple Graph Let V be the set of states in the far-southeastern U.S.: I.e., V={FL, GA, AL, MS, LA, SC, TN, NC} Let E={{u,v}|u adjoins v} ={{FL,GA},{FL,AL},{FL,MS}, {FL,LA},{GA,AL},{AL,MS}, {MS,LA},{GA,SC},{GA,TN}, {SC,NC},{NC,TN},{MS,TN}, {MS,AL}} NC TN MS AL SC GA LA FL 9/17/2018 (c)2001-2003, Michael P. Frank

Extensions All the main types of graphs can be extended to make them more expressive For example, edges may be labelled: Labelled graphs E.g., the edges in the previous example may be labelled with the type of border (e.g., patrolled/non-patrolled) 9/17/2018 (c)2001-2003, Michael P. Frank

Multigraphs Like simple graphs, but there may be more than one edge connecting two given nodes. A multigraph G=(V, E, f ) consists of a set V of vertices, a set E of edges (as primitive objects), and a function f:E{{u,v}|u,vV  uv}. E.g., nodes are cities, edges are segments of major highways. Parallel edges 9/17/2018 (c)2001-2003, Michael P. Frank

Multigraphs extended Extension: edges may be labelled with numbers of miles. How about this example? 110 A B 90 20 80 C 9/17/2018 (c)2001-2003, Michael P. Frank

Discrete Mathematics and its Applications 9/17/2018 Multigraphs extended Can go from A to B via C in less than 110. The distances in the graph are not minimal, or something is wrong. 110 A B Exercise: formulate a constraint on labels that prevents this kind of mishap. 90 20 80 C 9/17/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Pseudographs Like a multigraph, but edges connecting a node to itself are allowed. (R may even be reflexive.) A pseudograph G=(V, E, f ) where f:E{{u,v}|u,vV}. Edge eE is a loop if f(e)={u,u}={u}. E.g., nodes are campsites in a state park, edges are hiking trails through the woods. 9/17/2018 (c)2001-2003, Michael P. Frank

Directed Graphs Correspond to arbitrary binary relations R, which need not be symmetric. A di(rected) graph (V,E) consists of a set of vertices V and a binary relation E on V. E.g.: V = set of People, E={(x,y) | x loves y} 9/17/2018 (c)2001-2003, Michael P. Frank

Directed Multigraphs Like directed graphs, but there may be more than one edge from a node to another. A directed multigraph G=(V, E, f ) consists of a set V of vertices, a set E of edges, and a function f:EVV. E.g., V=web pages, E=hyperlinks. The WWW as a directed multigraph... 9/17/2018 (c)2001-2003, Michael P. Frank

Types of Graphs: Summary Summary of the book’s definitions. This terminology is not fully standardized across different authors 9/17/2018 (c)2001-2003, Michael P. Frank

§8.2: Graph Terminology Introducing the following terms: Adjacent, connects, endpoints, degree, initial, terminal, in-degree, out-degree, complete, cycles, wheels, n-cubes, bipartite, subgraph, union. 9/17/2018 (c)2001-2003, Michael P. Frank

Adjacency Let G be an undirected graph with edge set E. Let eE be (or map to) the pair {u,v}. Then we say: u, v are adjacent / connected. Edge e is incident with vertices u and v. Edge e connects u and v. Vertices u and v are endpoints of edge e. 9/17/2018 (c)2001-2003, Michael P. Frank

Degree of a Vertex Let G be an undirected graph, vV a vertex. The degree of v, deg(v), is its number of incident edges. (Except that any self-loops are counted twice.) A vertex with degree 0 is called isolated. A vertex of degree 1 is called pendant. 9/17/2018 (c)2001-2003, Michael P. Frank

Degree of a Vertex The degree of v, deg(v), is its number of incident edges. (Except that any self-loops are counted twice.) Exercise: construct a pseudograph with three vertices, all of which have different degrees. 9/17/2018 (c)2001-2003, Michael P. Frank

Handshaking Theorem Let G be an undirected (simple, multi-, or pseudo-) graph with vertex set V and edge set E. Then Proof: every edge causes degree:= degree +2 9/17/2018 (c)2001-2003, Michael P. Frank

Handshaking Theorem Let G be an undirected (simple, multi-, or pseudo-) graph with vertex set V and edge set E. Then Corollary: Any undirected graph has an even number of vertices of odd degree. 9/17/2018 (c)2001-2003, Michael P. Frank

`Sociological’ example (Like previous exercise, this is from Jonathan L.Gross’ coursenotes to Rosen) Suppose the students in this class are represented as vertices. An edge between a and b means that a and b were acquainted before the course began. [This is a simple graph.] How many students knew an odd number of other students? Could there be 3? 9/17/2018 (c)2001-2003, Michael P. Frank

`Sociological’ example (Like previous exercise, this is from Jonathan L.Gross’ coursenotes to Rosen) Suppose the students in this class are represented as vertices. An edge between a and b means that a and b were acquainted before the course began. [This is a simple graph.] [By corollary to handshaking theorem:] The number of students who knew an odd number of other students is even. 9/17/2018 (c)2001-2003, Michael P. Frank

Directed Adjacency Let G be a directed (possibly multi-) graph, and let e be an edge of G that is (or maps to) (u,v). Then we say: u is adjacent to v, v is adjacent from u e comes from u, e goes to v. e connects u to v, e goes from u to v the initial vertex of e is u the terminal vertex of e is v 9/17/2018 (c)2001-2003, Michael P. Frank

Directed Degree Let G be a directed graph, v a vertex of G. The in-degree of v, deg(v), is the number of edges going to v. The out-degree of v, deg(v), is the number of edges coming from v. The degree of v, deg(v):deg(v)+deg(v), is the sum of v’s in-degree and out-degree. 9/17/2018 (c)2001-2003, Michael P. Frank

Directed Handshaking Theorem Let G be a directed (possibly multi-) graph with vertex set V and edge set E. Then: Note that the degree of a node is unchanged by whether we consider its edges to be directed or undirected. 9/17/2018 (c)2001-2003, Michael P. Frank

Special Graph Structures Special cases of undirected graph structures: Complete graphs Kn Cycles Cn Wheels Wn n-Cubes Qn Bipartite graphs Complete bipartite graphs Km,n 9/17/2018 (c)2001-2003, Michael P. Frank

Complete Graphs For any nN, a complete graph on n vertices, Kn, is a simple graph with n nodes in which every node is adjacent to every other node: u,vV: uv{u,v}E. K1 K4 K2 K3 K5 K6 Note that Kn has edges. 9/17/2018 (c)2001-2003, Michael P. Frank

Consider a complete graph G=(V,E) Consider a complete graph G=(V,E). Can E contain any edges connecting a node in V to itself? 9/17/2018 (c)2001-2003, Michael P. Frank

No: this would mean {u,v}E, where u=v hence u,vV: uv{u,v}E. Consider a complete graph G=(V,E). Can E contain any edges connecting a node in V to itself? No: this would mean {u,v}E, where u=v hence u,vV: uv{u,v}E. 9/17/2018 (c)2001-2003, Michael P. Frank

Can you think of a natural example where complete graphs might model the facts? 9/17/2018 (c)2001-2003, Michael P. Frank

Can you think of a natural example where complete graphs might model the facts? E.g., Graphs that model, for groups of n people, which people are seen by each member of the group 9/17/2018 (c)2001-2003, Michael P. Frank

Discrete Mathematics and its Applications 9/17/2018 Cycles For any n3, a cycle on n vertices, Cn, is a simple graph where V={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}. Number of edges equals n. C3 C4 C5 C6 C8 C7 How many edges are there in Cn? 9/17/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Can you think of a natural example where cycles might model the facts? E.g., Graphs that model, for groups of n people, which people are sitting next to each other (on a round table) 9/17/2018 (c)2001-2003, Michael P. Frank

Can a cycle be a complete graph? 9/17/2018 (c)2001-2003, Michael P. Frank

Discrete Mathematics and its Applications 9/17/2018 Can a cycle be a complete graph? Yes: every cycle with exactly 3 elements is a complete graph. No other cycle can be a complete graph. Can be proven by proving with induction that |Edges(C_n)|<|Edges(K_n)| for n>2. 9/17/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Mathematics and its Applications 9/17/2018 Wheels For any n3, a wheel Wn, is a simple graph obtained by taking the cycle Cn and adding one extra vertex vhub and n extra edges {{vhub,v1}, {vhub,v2},…,{vhub,vn}}. The number is 2n W3 W4 W5 W6 W8 W7 How many edges are there in Wn? 9/17/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Definition: A graph is regular iff every vertex has the same degree Which of these are regular? (What degree?) Complete graphs? Cycle graphs? Wheel graphs? 9/17/2018 (c)2001-2003, Michael P. Frank

A graph is regular iff every vertex has the same degree Which of these are regular? (What degree?) Complete graphs? Yes: degree n-1 (for n nodes) Cycle graphs? Yes: degree 2 Wheel graphs? No, except when they have 3 nodes 9/17/2018 (c)2001-2003, Michael P. Frank

Number of vertices: 2n. Number of edges:Exercise to try! n-cubes (hypercubes) For any nN, the hypercube Qn is a simple graph consisting of two copies of Qn-1 connected together at corresponding nodes. Q0 has 1 node. Q0 Q1 Q2 Q4 Q3 Number of vertices: 2n. Number of edges:Exercise to try! 9/17/2018 (c)2001-2003, Michael P. Frank

n-cubes (hypercubes) For any nN, the hypercube Qn can be defined recursively as follows: Q0={{v0},} (one node and no edges) For any nN, if Qn=(V,E), where V={v1,…,va} and E={e1,…,eb}, then Qn+1 = … 9/17/2018 (c)2001-2003, Michael P. Frank

n-cubes (hypercubes) For any nN, the hypercube Qn can be defined recursively as follows: Q0={{v0},} (one node and no edges) For any nN, if Qn=(V,E), where V={v1,…,va} and E={e1,…,eb}, then Qn+1= (V{v1´,…,va´}, E{e1´,…,eb´}{{v1,v1´},{v2,v2´},…, {va,va´}}) where v1´,…,va´ are new vertices, and where if ei={vj,vk} then ei´={vj´,vk´}. 9/17/2018 (c)2001-2003, Michael P. Frank

Bipartite Graphs Def’n.: A graph G=(V,E) is bipartite (two-part) iff V = V1  V2 where V1∩V2= and eE: v1V1,v2V2: e={v1,v2}. In English: The graph can be divided into two parts in such a way that all edges go between the two parts. V1 V2 This definition can easily be adapted for the case of multigraphs and directed graphs as well. Can represent with zero-one matrices. 9/17/2018 (c)2001-2003, Michael P. Frank

Bipartite graphs … are extremely common, for example when you’re modelling a domain that consists of two different kinds of entities Animals in a zoo, linked with their keepers Words, linked with numbers of letters in them Logical formulas, linked with English sentences that express their meaning 9/17/2018 (c)2001-2003, Michael P. Frank

Some questions Can you think of a graph with two vertices that is not bipartite? Can you think of a simple graph with two vertices that is not bipartite? Can you think of a simple graph with three vertices and a nonempty set of edges that is bipartite? 9/17/2018 (c)2001-2003, Michael P. Frank

Some questions Can you think of a graph with two vertices that is not bipartite? Yes: if there are self-loops Can you think of a simple graph with two vertices that is not bipartite? No: must be bipartite Can you think of a simple graph with three vertices and a nonempty set of edges that is bipartite? Yes: as long as it’s not complete 9/17/2018 (c)2001-2003, Michael P. Frank

Given a (bipartite) graph, can there be more than 1 way of partitioning V into V1 and V2 ? 9/17/2018 (c)2001-2003, Michael P. Frank

Bipartite Graphs Given a (bipartite) graph, can there be more than 1 way of partitioning V into V1 and V2 ? Yes: isolated vertices can be put in either part: V1 V2 9/17/2018 (c)2001-2003, Michael P. Frank

Complete Bipartite Graphs Discrete Mathematics and its Applications 9/17/2018 Complete Bipartite Graphs For m,nN, the complete bipartite graph Km,n is a bipartite graph where |V1| = m, |V2| = n, and E = {{v1,v2}|v1V1  v2V2}. That is, there are m nodes in the left part, n nodes in the right part, and every node in the left part is connected to every node in the right part. K4,3 Note: not every CBG is complete! (So “CBG” is a strange name) K_{m,n} has m+n nodes and n*m edges. Km,n has _____ nodes and _____ edges. 9/17/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Subgraphs A subgraph of a graph G=(V,E) is a graph H=(W,F) where WV and FE. G H 9/17/2018 (c)2001-2003, Michael P. Frank

A subgraph of a graph G=(V,E) is a graph H=(W,F) where WV and FE. Note: since H is a graph, F can only involve nodes that are elements of W. 9/17/2018 (c)2001-2003, Michael P. Frank

§8.3: Graph Representations & Isomorphism Adjacency lists. Adjacency matrices. [Not covered this year] Incidence matrices. [Not covered this year] 9/17/2018 (c)2001-2003, Michael P. Frank

Adjacency Lists A table with 1 row per vertex, listing its adjacent vertices. b a d c e f 9/17/2018 (c)2001-2003, Michael P. Frank

An equivalence relation between graphs Unlike ordinary pictures, we can say precisely when two graphs are similar Obviously, a given graph (V,E) may be drawn in different ways. But even (V,E) and (V’,E’) (where V<>V’ and E<>E’) may in some sense be equivalent: Graph isomorphism (informal): Two graphs are isomorphic iff they are identical except for their node names. 9/17/2018 (c)2001-2003, Michael P. Frank

How would you define graph isomorphism formally? Graphs that are isomorphic share all their `important’ properties, e.g., The number of nodes and edges The degrees of all their nodes Whether they are bipartite or not, etc. How would you define graph isomorphism formally? For simplicity: focus on simple graphs Hint: use the notion of a bijection 9/17/2018 (c)2001-2003, Michael P. Frank

Graph Isomorphism Formal definition: Simple graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic iff  a bijection f:V1V2 such that  a,bV1, a and b are adjacent in G1 iff f(a) and f(b) are adjacent in G2. f is the “renaming” function between the two node sets that makes the two graphs identical. This definition can be extended to other types of graphs. 9/17/2018 (c)2001-2003, Michael P. Frank

Graph Isomorphism How can we tell whether two graphs are isomorphic? The best algorithms that are known to solve this problem have exponential worst-case time complexity. (Faster solutions may be possible.) In practice, a few tests go a long way … 9/17/2018 (c)2001-2003, Michael P. Frank

Graph Invariants under Isomorphism Necessary but not sufficient conditions for G1=(V1, E1) to be isomorphic to G2=(V2, E2): |V1|=|V2| and |E1|=|E2|. The number of vertices with degree n is the same in both graphs. For every proper subgraph g of one graph, there is a proper subgraph of the other graph that is isomorphic to g. 9/17/2018 (c)2001-2003, Michael P. Frank

Isomorphism Example If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. d b a b a d c e e c f f 9/17/2018 (c)2001-2003, Michael P. Frank

Are These Isomorphic? If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. Same # of vertices a b Same # of edges Different # of verts of degree 2! (1 vs 3) d e c 9/17/2018 (c)2001-2003, Michael P. Frank

We now jump to the area that gave raise to the invention of (a formal theory of) graphs Questions like “Can I travel from a to b?” “Can I go travel a to b without going anywhere twice?” “What’s the quickest route from a to b?” 9/17/2018 (c)2001-2003, Michael P. Frank

§8.4: Connectivity In an undirected graph, a path of length n from u to v is a sequence of adjacent edges going from vertex u to vertex v. A path is a circuit if u=v. A path traverses the vertices along it. A path is simple if it contains no edge more than once. 9/17/2018 (c)2001-2003, Michael P. Frank

Connectedness An undirected graph is connected iff there is a path between every pair of distinct vertices in the graph. Theorem: There is a simple path between any pair of vertices in a connected undirected graph. 9/17/2018 (c)2001-2003, Michael P. Frank

( Directed Connectedness A directed graph is strongly connected iff there is a directed path from a to b for any two verts a and b. It is weakly connected iff the underlying undirected graph (i.e., with edge directions removed) is connected. ) 9/17/2018 (c)2001-2003, Michael P. Frank

Paths & Isomorphism Note that connectedness, and the existence of a circuit or simple circuit of length k are graph invariants with respect to isomorphism. 9/17/2018 (c)2001-2003, Michael P. Frank

§8.5: Euler & Hamilton Paths We’ll show you the problem that prompted Euler to invent the theory of graphs: the bridges of Koenigsberg (town later called Kaliningrad) 9/17/2018 (c)2001-2003, Michael P. Frank

Bridges of Königsberg Problem Can we walk through town, crossing each bridge exactly once, and return to start? 9/17/2018 (c)2001-2003, Michael P. Frank

Bridges of Königsberg Problem Can we walk through town, crossing each bridge exactly once, and return to start? A Can you model the situation using a graph? D B C The original problem 9/17/2018 (c)2001-2003, Michael P. Frank

Bridges of Königsberg Problem Can we walk through town, crossing each bridge exactly once, and return to start? A D B C Equivalent multigraph The original problem 9/17/2018 (c)2001-2003, Michael P. Frank

§8.5: Euler & Hamilton Paths Terminology: An Euler circuit in a graph G is a simple circuit containing every edge of G. An Euler path in G is a simple path containing every edge of G. 9/17/2018 (c)2001-2003, Michael P. Frank

Bridges of Koenigsberg Bridges are edges. So the answer to the problem is YES iff its graph contains an Euler circuit. In fact, it does not … 9/17/2018 (c)2001-2003, Michael P. Frank

Euler Path Theorems Theorem: A finite connected multigraph has an Euler circuit iff each vertex has even degree. Proof: (→) The circuit contributes 2 to degree of each node. (←) By construction using algorithm on p. 580-581 Theorem: A connected multigraph has an Euler path iff it has exactly 2 vertices of odd degree. One is the start, the other is the end. 9/17/2018 (c)2001-2003, Michael P. Frank

Not all edges have even degree … so there is no Euler circuit. A D B C Equivalent multigraph The original problem 9/17/2018 (c)2001-2003, Michael P. Frank

Euler Circuit theorem Sketch of proof that even degree implies existence of Euler circuit: Start with any arbitrary node. Construct simple path from it till you get back to start. (Graph is connected; every node has even degree, so you can leave any node that you have entered) Repeat for each remaining subgraph, splicing results back into original cycle. Finiteness of graph implies that this process must end. Note that the complete version of this proof provides an algorithm: it’s a constructive proof of an existential proposition 9/17/2018 (c)2001-2003, Michael P. Frank

( Hamilton Paths An Euler circuit in a graph G is a simple circuit containing every edge of G. An Euler path in G is a simple path containing every edge of G. A Hamilton circuit is a circuit that traverses each vertex in G exactly once. A Hamilton path is a path that traverses each vertex in G exactly once. ) 9/17/2018 (c)2001-2003, Michael P. Frank

( Hamiltonian Path Theorems Dirac’s theorem: If (but not only if) G is connected, simple, has n3 vertices, and v deg(v)n/2, then G has a Hamilton circuit. Ore’s corollary: If G is connected, simple, has n≥3 nodes, and deg(u)+deg(v)≥n for every pair u,v of non-adjacent nodes, then G has a Hamilton circuit. ) 9/17/2018 (c)2001-2003, Michael P. Frank

( HAM-CIRCUIT is NP-complete Let HAM-CIRCUIT be the problem: Given a simple graph G, does G contain a Hamiltonian circuit? This problem has been proven to be NP-complete! This means, if an algorithm for solving it in polynomial time were found, it could be used to solve all NP problems in polynomial time. ) 9/17/2018 (c)2001-2003, Michael P. Frank

§8.6: Shortest-Path Problems Not covering this year. 9/17/2018 (c)2001-2003, Michael P. Frank

§8.7: Planar Graphs Not covering this year. 9/17/2018 (c)2001-2003, Michael P. Frank

§9.1: Introduction to Trees A tree is a connected undirected graph that contains no circuits. Theorem: There is a unique simple path between any two of its nodes. A (not-necessarily-connected) undirected graph without simple circuits is called a forest. You can think of it as a set of trees having disjoint sets of nodes. A leaf node in a tree or forest is any pendant or isolated vertex. An internal node is any non-leaf vertex (thus it has degree ≥ ___ ). 9/17/2018 (c)2001-2003, Michael P. Frank

Tree and Forest Examples Leaves in green, internal nodes in brown. A Tree: A Forest: 9/17/2018 (c)2001-2003, Michael P. Frank

Rooted Trees A rooted tree is a tree in which one node has been designated the root. Every edge is (implicitly or explicitly) directed away from the root. Concepts related to rooted trees: Parent, child, siblings, ancestors, descendents, leaf, internal node, subtree. 9/17/2018 (c)2001-2003, Michael P. Frank

Same tree except for choice of root Rooted Tree Examples Note that a given unrooted tree with n nodes yields n different rooted trees. Same tree except for choice of root root root 9/17/2018 (c)2001-2003, Michael P. Frank

Rooted-Tree Terminology Exercise Find the parent, children, siblings, ancestors, & descendants of node f. o n h r d m b root a c g e q i f l p j k 9/17/2018 (c)2001-2003, Michael P. Frank

n-ary trees A rooted tree is called n-ary if every vertex has no more than n children. It is called full if every internal (non-leaf) vertex has exactly n children. A 2-ary tree is called a binary tree. These are handy for describing sequences of yes-no decisions. Example: Comparisons in binary search algorithm. 9/17/2018 (c)2001-2003, Michael P. Frank

Which Tree is Binary? Theorem: A given rooted tree is a binary tree iff every node other than the root has degree ≤ ___, and the root has degree ≤ ___. 9/17/2018 (c)2001-2003, Michael P. Frank

Ordered Rooted Tree This is just a rooted tree in which the children of each internal node are ordered. In ordered binary trees, we can define: left child, right child left subtree, right subtree For n-ary trees with n>2, can use terms like “leftmost”, “rightmost,” etc. 9/17/2018 (c)2001-2003, Michael P. Frank

Trees as Models Can use trees to model the following: Saturated hydrocarbons Organizational structures Computer file systems In each case, would you use a rooted or a non-rooted tree? 9/17/2018 (c)2001-2003, Michael P. Frank

Some Tree Theorems Any tree with n nodes has e = n−1 edges. Proof: Consider removing leaves. A full m-ary tree with i internal nodes has n=mi+1 nodes, and =(m−1)i+1 leaves. Proof: There are mi children of internal nodes, plus the root. And,  = n−i = (m−1)i+1. □ Thus, when m is known and the tree is full, we can compute all four of the values e, i, n, and , given any one of them. 9/17/2018 (c)2001-2003, Michael P. Frank

Some More Tree Theorems Definition: The level of a node is the length of the simple path from the root to the node. The height of a tree is maximum node level. A rooted m-ary tree with height h is called balanced if all leaves are at levels h or h−1. Theorem: There are at most mh leaves in an m-ary tree of height h. Corollary: An m-ary tree with  leaves has height h≥logm . If m is full and balanced then h=logm. 9/17/2018 (c)2001-2003, Michael P. Frank

§9.2: Applications of Trees Binary search trees A simple data structure for sorted lists Decision trees Minimum comparisons in sorting algorithms Prefix codes Huffman coding Game trees 9/17/2018 (c)2001-2003, Michael P. Frank

Binary Search Trees A representation for sorted sets of items. Supports the following operations in Θ(log n) average-case time: Searching for an existing item. Inserting a new item, if not already present. Supports printing out all items in Θ(n) time. Note that inserting into a plain sequence ai would instead take Θ(n) worst-case time. 9/17/2018 (c)2001-2003, Michael P. Frank

Binary Search Tree Format Items are stored at individual tree nodes. We arrange for the tree to always obey this invariant: For every item x, Every node in x’s left subtree is less than x. Every node in x’s right subtree is greater than x. Example: 7 3 12 1 5 9 15 2 8 11 9/17/2018 (c)2001-2003, Michael P. Frank

Recursive Binary Tree Insert procedure insert(T: binary tree, x: item) v := root[T] if v = null then begin root[T] := x; return “Done” end else if v = x return “Already present” else if x < v then return insert(leftSubtree[T], x) else {must be x > v} return insert(rightSubtree[T], x) 9/17/2018 (c)2001-2003, Michael P. Frank

Decision Trees (pp. 646-649) A decision tree represents a decision-making process. Each possible “decision point” or situation is represented by a node. Each possible choice that could be made at that decision point is represented by an edge to a child node. In the extended decision trees used in decision analysis, we also include nodes that represent random events and their outcomes. 9/17/2018 (c)2001-2003, Michael P. Frank

Coin-Weighing Problem Imagine you have 8 coins, one of which is a lighter counterfeit, and a free-beam balance. No scale of weight markings is required for this problem! How many weighings are needed to guarantee that the counterfeit coin will be found? ? 9/17/2018 (c)2001-2003, Michael P. Frank

As a Decision-Tree Problem In each situation, we pick two disjoint and equal-size subsets of coins to put on the scale. A given sequence of weighings thus yields a decision tree with branching factor 3. The balance then “decides” whether to tip left, tip right, or stay balanced. 9/17/2018 (c)2001-2003, Michael P. Frank

Applying the Tree Height Theorem The decision tree must have at least 8 leaf nodes, since there are 8 possible outcomes. In terms of which coin is the counterfeit one. Recall the tree-height theorem, h≥logm. Thus the decision tree must have height h ≥ log38 = 1.893… = 2. Let’s see if we solve the problem with only 2 weighings… 9/17/2018 (c)2001-2003, Michael P. Frank

General Solution Strategy The problem is an example of searching for 1 unique particular item, from among a list of n otherwise identical items. Somewhat analogous to the adage of “searching for a needle in haystack.” Armed with our balance, we can attack the problem using a divide-and-conquer strategy, like what’s done in binary search. We want to narrow down the set of possible locations where the desired item (coin) could be found down from n to just 1, in a logarithmic fashion. Each weighing has 3 possible outcomes. Thus, we should use it to partition the search space into 3 pieces that are as close to equal-sized as possible. This strategy will lead to the minimum possible worst-case number of weighings required. 9/17/2018 (c)2001-2003, Michael P. Frank

General Balance Strategy On each step, put n/3 of the n coins to be searched on each side of the scale. If the scale tips to the left, then: The lightweight fake is in the right set of n/3 ≈ n/3 coins. If the scale tips to the right, then: The lightweight fake is in the left set of n/3 ≈ n/3 coins. If the scale stays balanced, then: The fake is in the remaining set of n − 2n/3 ≈ n/3 coins that were not weighed! Except if n mod 3 = 1 then we can do a little better by weighing n/3 of the coins on each side. You can prove that this strategy always leads to a balanced 3-ary tree. 9/17/2018 (c)2001-2003, Michael P. Frank

Coin Balancing Decision Tree Here’s what the tree looks like in our case: 123 vs 456 left: 123 balanced: 78 right: 456 1 vs. 2 4 vs. 5 7 vs. 8 L:1 R:2 B:3 L:4 R:5 B:6 L:7 R:8 9/17/2018 (c)2001-2003, Michael P. Frank

Prefix Codes & Huffman Coding pp. 649-651 9/17/2018 (c)2001-2003, Michael P. Frank

Game Trees pp. 651-656 9/17/2018 (c)2001-2003, Michael P. Frank

§9.3: Tree Traversal Universal address systems Traversal algorithms Depth-first traversal: Preorder traversal Inorder traversal Postorder traversal Breadth-first traversal Infix/prefix/postfix notation 9/17/2018 (c)2001-2003, Michael P. Frank

§9.4: Spanning Trees Not covering this semester. 9/17/2018 (c)2001-2003, Michael P. Frank

§9.5: Minimum Spanning Trees Not covering this semester. 9/17/2018 (c)2001-2003, Michael P. Frank